London's population is already at a post-war high of 8.3 million and is predicted to grow by another 700,000 in just six years. That is the equivalent of squeezing in two entire London boroughs.

Such dramatic population growth raises a multitude of questions for policymakers at all levels – where will the extra schools places be found, how will the NHS meet the needs of a growing population, but above all where are those people going to live?

At London Councils we have been looking at the housing challenge in particular. Our analysis highlights a huge backlog to deal with even before we start creating the news homes for that rising population. We need to build more than 800,000 new homes in London by 2021 to meet new and existing housing demand.

The figures are stark and demonstrate just how big the challenge is. House prices in London have risen 389% since 1983, while household income has risen by just 159% in the same period. The average rent for a three-bedroom flat is £1,833 per month, and consumes 59% of a London family's income. Londoners now experience the highest levels of overcrowding in the country and the capital has more than 41,250 homeless households in temporary accommodation.

There cannot be a single solution to a crisis of this severity. Focusing our energy on just building council homes, or relying on housing associations and private developers to build the homes we need will not be enough. We need every single type of housebuilder to deliver more. And the new homes we build can't just be of one tenure type, we need homes that meet the needs of every Londoner. We need homes for social rent, private rent and outright sale.

Our report sets out some ideas for tackling the problems we face. Removing the housing revenue account borrowing cap would allow councils to borrow against their housing stock and build homes themselves. This could allow an extra £2.5bn borrowing capacity to be released, enabling councils in London to build an extra 13,900 homes by 2021.

Introducing an undeveloped land tax could help to reduce land banking and encourage housebuilders to start work on the 124,247 homes with planning permission in London but which have not been built.

But perhaps the biggest prize is to move from a system of personal housing subsidy, through housing benefit, to investing in bricks and mortar. Over the past 10 years £50bn has been spent on housing benefit in London, more than three times the £17bn of capital investment in building new homes and improving existing ones.

The Institute of Public Policy Research wrote about this in detail last year, but one way to achieve it would be for government to develop an earn-back deal that would give local authorities a share of the savings made from reducing the overall housing benefit bill to be recycled into building more homes at lower rents.

The housing shortage is the biggest single issue facing our city. If we do not find solutions to it we risk London's future. Politicians of all parties need to work together and be willing to thing differently, think fast and think big.